Hexagonal cylinder

Fig-1 Mean-field predication of the morphologies for conformationally symmetric diblock melts. Phases are labeled as S (bcc spheres), C (hexagonal cylinders), G (bicontin-uous la 3 d. cubic), L (lamellar)./a is the volume fraction... 

Fig. 9 Schematic representation of three approaches to generate nanoporous and meso-porous materials with block copolymers, a Block copolymer micelle templating for mesoporous inorganic materials. Block copolymer micelles form a hexagonal array. Silicate species then occupy the spaces between the cylinders. The final removal of micelle template leaves hollow cylinders, b Block copolymer matrix for nanoporous materials. Block copolymers form hexagonal cylinder phase in bulk or thin film state. Subsequent crosslinking fixes the matrix hollow channels are generated by removing the minor phase, c Rod-coil block copolymer for microporous materials. Solution-cast micellar films consisted of multilayers of hexagonally ordered arrays of spherical holes. (Adapted from [33])...

Fig. 2.26 TEM from a poly(2-vinylpyridine)-poly(isoprene)- poly(styrene) ABC triblock copolymer, showing a hexagonal cylinder core-shell structure (Gidcma/. 1993).The light, grey and black regions correspond to the PS, P2VP and PI respectively in sections (a) perpendicular and (b) parallel to the rod direction.

The experimental values of FK and Fz can be interpreted by simple geometric considerations. A cross section of a hexagonal cylinder packing with the volume fraction 8 = 0.25 shows, for example, that the shells... 

Fig. 63 X-shaped quaternary bolaamphiphile 189 forming hexagonal cylinder phases with models showing (a) the three-color, (b) two-color and (c) single color hexagon tiling patterns in the left bottom comer the core-shell structure of the mixed cells is illustrated dark = Rp-chains white = Rsi-chains gray = mixed cells [42]...

Figure 21.19 SAXS traces with mesophase assignment of the polymer—inorganic composite films and SEM images of individual nanoparticles (a) sphere, (b) hexagonal cylinder, and (c) lamellar morphologies. The schematic in the middle shows nanoparticle shapes obtained from each phase through dissolution of the films and subsequent calcinations.61 (Reprinted with permission from C. B. W. Garcia et al., J. Am. Chem. Soc. 2003, 125, 13310-13311. Copyright 2003 American Chemical Society.)...

Exotic phases are found at compositions between lamellae and hexagonal cylinders (see Figs. 12-19 and 12-20). Some examples of the morphologies of these phases are shown in Fig. 12-22 these include cubic strut phases, tetragonal and rhombohedral mesh phases, and rectangular ribbon phases. 

Figure 13.4 Phase diagram of a PS-PI diblock copolymer showing regions of BCC spheres (Im3m), hexagonal cylinders (HEX), laid gyroid, hexagonally perforated lamellae (HPL), lamellar (LAM), and disordered phases /pi is the volume fraction of polyisoprene. The dot-dash line represents the mean-field order-disorder transition based on the formula x = 71.4/T — 0.0857 with reference segment volume v — 144 A. (Reprinted with permission from Khandpur et al.. Macromolecules 28 8796. Copyright 1995, American Chemical Society.)...

As with lamellae-forming samples, when the hexagonal phase forms after a quench from the disordered state, terminal behavior in G and G" is replaced by nonterminal behavior (see Fig. 13-13). For hexagonal cylinders, the low-frequency behavior is generally more elastic than for lamellae the apparent power-law exponent in G oc tu is usually somewhat lower (n 0.0-0.4) for the former than for the latter (n 0.5) (Gouinlock and Porter 1977 Morrison et al. 1990 Winter et al. 1993 Almdal et al. 1992). This is perhaps not surprising, since ordered hexagonal cylinders are solid-like (i.c., positionally ordered) in two dimensions and lamellae are positionally ordered in only one. 

Preliminary calculations suggest that copol)rmer melts in the presence of added homopol)rmer e.g. pure A or B polymer) should form bicontinuous "sponge" and "mesh" structures between lamellar and hexagonal (cylinder) phases. These geometries have indeed been observed [48]. The example shown in Fig. 4.27 illustrates the difficulty in distinguishing these hyperbolic phases from "classical" phases, mentioned in section 4.7. The mesh... 

Fig. 3. Computer simulation results using a time-dependent Ginzburg-Landau approach, showing the microstructural evolution after a temperature jump from the lamellar phase to the hexagonal cylinder phase for a moderately asymmetric diblock copolymer. The time units are arbitrary. (Reprinted with permission from Polymer 39, S. Y. Qi and Z.-G. Zheng, Weakly segregated block copolymers Anisotropic fluctuations and kinetics of order-order and order-disorder transitions, 4639-4648, copyright 1998, with permission of Excerpta Medica Inc.)...

Figure 13.13 Mean-field phase diagram tor confoimationally symmetric diblock copolymer melts. Phases are labeled L, lamellar H, hexagonal cylinders Qia3d, bicontinuous Ia3d cubic Qimsm, bcc spheres CPS, close-packed spheres and DIS, disordered region. The dashed lines denote extrapolated phase boundaries, and the dot denotes the mean-field critical ODT.

Figure 28.2 Basic structure of the calcium fluorapatite lattice. The large hexagonal cylinders have been formed by joining the centres of Ca " ions in the crystal lattice. The projection of a phosphate tetrahedron is a triangle...

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